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## Definition: "Lies on" Relation

Let $\mathcal P$, $\mathcal L$ and $\Pi$ be sets of points, straight lines, and planes respectively.

The “lies on” relation $R$ is an arbitrary relation $R\subseteq \mathcal P\times\mathcal L.$ We say that a point $A\in \mathcal P$ lies on a straight line $h\in \mathcal L$ if the ordered pair $(A,h)$ is contained in $R$ in set-theoretical sense, i.e. $(A,h)\in R.$

Similarly, the “lies on” relation $S\subseteq \mathcal P\times\Pi$ means that a point $A\in \mathcal P$ lies on a plane $\alpha \in \Pi$ if the ordered pair $(A,\alpha)\in S$ in set-theoretical sense.

### Notes

• In undergraduate mathematics, many authors define the “lies on” relation by the contained relation. A point $A$ lies on $h$ if it is contained in $h$, i.e. $A\in h.$ Similarly, if $A\in\alpha$ then the point $A$ “lies on” the plane $\alpha.$ These definitions are the most descriptive and intuitive ones but not the only possible.
• The reader should be aware that in Hilbert’s original, general sense, the relations $R$ and $S$ are arbitrary. All that counts is the set-theoretical “being contained” of the ordered pairs $(A,h)\in R,$ respectively $(A,\alpha)\in S.$ Nevertheless, Hilbert called his relation “lies upon” for the sake of a better intuition.

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